6.2-04 Preserving lengths and angles

Preserving lengths and angles

Length and angle preservation can be derived directly from an orthogonal matrix identity. These questions connect that proof to distances, reflections, translations and the limitations of testing only a few vector lengths.

What you will practise

• Explain norm, dot-product and angle preservation.
• Use distances between transformed points.
• Distinguish linear isometries from translations.
• Recover cross-products from squared lengths.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review orthogonal matrices and Euclidean geometry. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports understanding which geometric features orthogonal transformations preserve within Module 6 of the Complete Linear Algebra Course.